Hi Jason!
Determining transportation times when traveling throughout Walt Disney World is tricky as traffic can be different at various times of the day and year. It is always recommended you give yourself at least an hour and a half to get from one place to the next just in case there is a traffic issue of any sort.
If you need to get from Hollywood Studios to the
Trails End Restaurant at the Fort Wilderness Campground you will simply need to find the Fort Wilderness bus at the park. Once you arrive at the Fort you will need to get off and hop on the internal bus. Let the driver know that you need to go to the restaurant located at Pioneer Hall. This trip could easily take an hour or more since you have two buses to catch.
When coming from the Magic Kingdom to Fort Wilderness for the
Mickey's Backyard BBQ you will need to catch the resort boat. This is located directly in front of the entrance to the park and is very convenient. The boats will run about every 20 minutes and will drop you off just steps away from the Barbecue location. Give yourself at least 45 minutes to make this trip.
Your final question is about transportation from Epcot to the Contemporary Resort where
Chef Mickey's is located. You can either take a bus to the resort or take the monorail. If you take the monorail you will need to transfer to the resort line at the Transportation and Ticket Center and then go all around the loop before stopping at the Contemporary. This would be the longest and I would give yourself two hours. By bus plan on about 45 minutes.